This paper presents an exact solution to the Timoshenko beam theory (TBT) for bending, second-order analysis, and stability. The TBT covers cases associated with small deflections based on shear deformation considerations, whereas the Euler–Bernoulli beam theory neglects shear deformations. A material law (a moment-shear force-curvature equation) combining bending and shear is presented, together with closed-form solutions based on this material law. A bending analysis of a Timoshenko beam was conducted, and buckling loads were determined on the basis of the bending shear factor. First-order element stiffness matrices were calculated. Finally second-order element stiffness matrices were deduced on the basis of the same principle.

There is need to accurately design pipelines to transport the expected increase of CO₂ captured from industrial processes after the signing of the Paris Climate Agreement in 2016. This paper reviews several aspects of CO₂ pipeline design with emphasis on pressure drop and models for the calculation of pipeline diameter. Two categories of pipeline equations were identified. The first category is independent of pipeline length and has two different equations. This category is used to specify adequate pipeline diameter for the volume of fluid transported. The optimum economic pipe diameter equation (Eq. 17) with nearly uniform resultant velocity values at different flow rates performed better than the standard velocity flow equation (Eq. 20). The second category has four different equations and is used to calculate pipeline pressure drop or pipeline distance for the installation of booster stations after specifying minimum and maximum pipeline pressures. The hydraulic equation is preferred because it gave better resultant velocity values and the closest diameter value obtained using Aspen HYSYS (V.10) simulation. The effect of impurities on the pressure behaviour and optimal pipeline diameter and pressure loss due to acceleration were ignored in the development of the models. Further work is ongoing to incorporate these effects into the models.

We introduce the Symplectic Structure of Information Geometry based on Souriau’s Lie Group Thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using Geometric (Planck) Temperature of Souriau model and Symplectic cocycle notion, the Fisher metric is identified as a Souriau Geometric Heat Capacity. Souriau model is based on affine representation of Lie Group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. In the framework of Lie Group Thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincaré-Cartan-Souriau integral. The Souriau-Fisher metric is linked to KKS (Kostant-Kirillov-Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We apply this model in the framework of Information Geometry for the action of an affine Group for exponentiel families, and provide some illustrations of use cases for multivariate Gaussian densities. Information Geometry is presented in the context of seminal work of Fréchet and his Clairaut-Legendre equation. Souriau model of Statistical Physics is validated as compatible with Balian gauge model of thermodynamics. We recall the precursor work of Casalis on affine group invariance for natural exponential families.

In the paper, the authors consider the generating functions of the Hermite polynomials and their squares, present explicit formulas for higher order derivatives of the generating functions of the Hermite polynomials and their squares, which can be viewed as ordinary differential equations or derivative polynomials, find differential equations that the generating functions of the Hermite polynomials and their squares satisfy, and derive explicit formulas and recurrence relations for the Hermite polynomials and their squares.

The paper presents an adaptation of numerical solution of first order linear differential equation in fuzzy environment. The numerical method is re-established and studied with fuzzy concept to estimate its uncertain parameters whose values are not precisely known. Demonstrations of fuzzy solutions of the governing methods are carried out by the approaches, namely Modified Runge Kutta method and Runge Kutta Merson method. The results are compared with the exact solution which is found using generalized Hukuhara derivative (gH-derivative) concepts. Additionally, different illustrative examples and an application in industry of the methods are also undertaken with the useful table and graph to show the usefulness for attained to the proposed approaches.

Background: Tourism is one of the prominent industry that has a capability to generate handsome income for developed as well as developing countries. However, the study to investigate specifically the perception of local residents towards tourism is still lacking. The perception of the locals is important since it could determine the extent of their support for tourism development. Plus, previous research found that male residents are more support better towards the tourism development. Thus, this factor is adapted as well in this study to examine whether this gender may influence to the tourism industry. 2) Methods: This study used Structural Equation Modeling (SEM) technique for determining the structural estimates between constructs. 3) Results: The respondents agree that positive perception, negative perception and tourism impact has a significant impact on support for tourism development in which has been recognized as Social Exchange Theory model. Moreover, the finding also revealed that gender has a potential to moderates the causal effect of tourism impact on tourism development. 4) Conclusion: The resident understand the tourism sector could strengthen economic nation but they also want the natural resources should be protected.

A solution for the Prandtl-Blasius equation is essential to all kinds of boundary layer problems. This paper revisits this classic problem and presents a general Maple code as its numerical solution. The solutions were obtained from the Maple code, using the Runge-Kutta method. The study also considers convergence radius expanding and an approximate analytic solution is proposed by curve fitting. Similarly, the study resolves some boundary layer related problems and provide relevant Maple codes for these.

Finding a solution for Euler's equations is a classic mechanics problem. This study revisits the problem with numerical approaches. For ease of teaching and research, a Maple code comprising 2 lines is written to find a numerical solution for the problem. The study's results are validated by comparing these with previous studies. Our results confirm the correctness of the principle of maximum moment of inertia of the rotating body, which is verified by thermodynamics. As an essential part of this study, the Maple code is provided.

Background: This study aimed to identify the association between occupational stress and depression-well-being by proposing a comprehensive and flexible job burden-capital model with its corresponding hypotheses. Methods: For this research, 1618 valid samples were gathered from the electronic manufacturing service industry in Hunan Province, China; self-rated questionnaires were administered to participants for data collection after obtaining their written consent. The proposed model was fitted and tested through structural equation model analysis. Results: Single-factor correlation analysis results indicated that coefficients between all items and dimensions had statistical significance. The final model demonstrated satisfactory global goodness of fit (CMIN/DF=5.37, AGFI=0.915, NNFI=0.945, IFI=0.952, RMSEA=0.052). Both the measurement and structural models showed acceptable path loadings. Job burden and capital were directly associated with depression and well-being or indirectly related to them through personality. Multi-group structural equation model analyses indicated general applicability of the proposed model to basic features of such a population. Gender, marriage and education led to differences in the relation between occupational stress and health outcomes. Conclusions: The job burden-capital model of occupational stress-depression and well-being was found to be more systematic and comprehensive than previous models.

In the present study, we derive a new B-spline technique namely trigonometric B-spline collocation algorithm to solve some initial boundary value problems for the nonlinear Klein-Gordon equation. In order to carry out the time integration with Crank-Nicolson implicit method, the order of the equation is reduced to give a coupled system of nonlinear partial differential equations. The collocation approximation based on trigonometric cubic B-splines for spatial discretization is followed by the linearization of the nonlinear term. The efficiency and accuracy of the present method are validated by measuring the error between the numerical and analytical solutions when exist. The conservation laws representing momentum and energy are also computed for all problems.

Rare earths are classified as most important and critical material for US economy and defense by Congress and a mandate has been set to increase their in-house production, domestic resource utilization and decrease reliance on foreign resources and reserves. They are widely available in earth crust as ore (bastnaesite (La, Ce)FCO₃, monazite, (Ce, La, Y, Th)PO₄, and xenotime, YPO₄), but their so-called economic reserves are sparsely located geographically. They may be produced by various means such as beneficiation (physical, chemical, mechanical, or electrical), reduction (direct or indirect), electrolysis (of aqueous or molten / fused single or mixed salt systems) at high temperature or hydrometallurgy. Out of these, direct reduction also known as metallothermic reduction (La and Ca reduction) is mostly utilized. Its variant, high temperature electrowinning of fused salts is also practiced widely. These processes are material and application specific. In this study, author will employ thermodynamics (Ellingham diagrams, free energy of formation, reduction potential, Nernst equation, Pourbaix (E_h-pH) diagrams, E-pO^-2 diagrams), kinetics and energetic of a chemical reaction (chemical metallurgy) to reduce rare earth oxide / salt to rare earth metals (REO/RES – REM). It is shown that materials and energy requirement vary greatly depending on type of mineral ore, production facility, and beneficiation / mineral processing method selected. Aim is to reduce dependence on coal deposits. It is anticipated this route will be able to produce rare earths with > 35% yield and > 98% purity which be described in subsequent studies and patents.

In this paper, we introduce some basic idea of Variational iteration method for short (VIM) to solve the Volterra’s integro-differential equations. The VIM is used to solve effectively, easily, and accurately a large class of non-linear problems with approximations which converge rapidly to accurate solutions. For linear problems, it’s exact solution can be obtained by only one iteration step due to the fact that the Lagrange multiplier can be exactly identified. It is to be noted that the Lagrange multiplier reduces the iteration on integral operator and also minimizes the computational time. The method requires no transformation or linearization of any forms. Two numerical examples are presented to show the effectiveness and efficiency of the method. Also, we compare the result with the result from Homotopy perturbation method (HPM). Finally, we investigate the absolute difference between variational iteration method and homotopy perturbation method and draw the graph of difference function by using Mathematica.

In this paper after introducing a model of binary data matrix (BDM) for physical parameters of an evolving system (of particles), we develop a Hilbert space as an ambient space to derive induced metric tensor on embedded parametric manifold identified by associated joint probabilities of particles observables (parameters). Parameter manifold assumed as space-like hypersurface evolving along time axis, an approach that resembles 3+1 formalism of ADM and numerical relativity. We show the relation of endowed metric with related density matrix. Identification of system density matrix by this metric tensor, leads to the equivalence of quantum Liouville equation and metric compatibility condition while covariant derivative of metric tensor has been calculated respect to Wick rotated time or spatial coordinates. After deriving a formula for expected energy per particles, we prove the equality of this expected energy with local scalar curvature of related manifold. We show the compatibility of BDM model with Hamilton-Jacobi formalism and canonical forms. On the basis of the model, I derive the Ricci flow like dynamics as the governing dynamics and subsequently derive the action of BDM model and Einstein field equations. Given examples clarify the compatibility of the results with well-known principles such as equipartition energy principle and Landauer’s principle. This model provides a background for geometrization of quantum mechanics compatible with curved manifolds and information geometry. Finally, we conclude a “bit density principle” which predicts the Planck equation, De Broglie wave particle relation, , Beckenstein bound and Bremermann limit.

The role of human resource management practices and organizational loyalty dealing with employee retention has been widely explored and examined in the existing literature. The majority of human resource management researchers have focused on the role of policies and practices in different sectors related to employee retention. However, less attention has been given to identifying the practices and procedures common across all industries and sectors. The present study attempts to determine and examine the most important constructs like HR practices and organizational loyalty of employee retention management and proposes a comprehensive structural equation model to measure the impact of these constructs on employee retention. After an extensive review of the literature, organizational loyalty, and human resources management practices like recruitment and selection, compensation and benefit, training and development, supervision and evaluation have been identified as four essential practices dealing with employee retention and are common across all industries. These practices constitute the component factors of an employee retention management predicting employee’s turnover intentions. The survey method was adopted, and the original data obtained through the self-administered questionnaire. The valid one hundred and eighty-nine (189) respondents analyzed by using Lisrel 8.7, Structural equation model (SEM) analysis results shows that human resource management practices and organizational loyalty have strong positive impact on employee retention. This study tends to assist human resource managers and decision makers in selecting the appropriate motivating factors to retain and satisfy their employees

Identification of costs drivers and their influence level on building development costs play a key role in the development of construction models and improve the efficiency and effectiveness of any project. Forty-five indicators influencing building development costs in New Zealand are explored by literature review and pilot interviews. These indicators are grouped into seven categories. The determination and ranking of the cost drivers are carried out by a questionnaire survey distributed to key professionals working in New Zealand’s construction industry. Structural equation modeling (SEM) software was employed for analysis of the collected data. One of the key advantages of this powerful software is to provide the p-value according to the structure of the research model. Findings of this study indicate that the property market and construction industry factor, statutory and regulatory factor, and socio-economic factor are major factors affecting building development costs in New Zealand.

Understory plants are important components of forest ecosystem productivity and diversity. Compared to biomass models of overstory canopy trees, few are available for understory saplings and shrubs and therefore their roles in estimation of forest carbon pools are often ignored. In this study, we harvested 24 understory species including 4 saplings, 9 tree-like shrubs and 11 typical shrubs in coniferous and broadleaved mixed forest in northeastern China and developed the best fit allometric equations of above- and below-ground and total biomass by species-specific or multispecies using morphological measurements of basal diameter, height and crown area as independent variables. The result showed that single basal diameter, height or crown area had good explanatory power for both species-specific and multispecies (p<0.001). The best-fit models included only basal diameter in sapling and tree-like shrubs, and combinations of crown area, height, and basal diameter in typical shrubs. The logarithmic model was most desired among the 4 model forms of linear, quadratic, multiple linear and logarithmic, for species-specific and multispecies. The models we developed should help the estimation of forest ecosystem carbon stocks, especially for belowground component, and provide tools for quantification of individual species biomass of understory plants.

Exact solutions to conformable time fractional (3+1)- dimensional equations are derived by using the modified form of the Kudryashov method. The compatible wave transformation reduces the equations to an ODE with integer orders. The predicted solution of the finite series of a rational exponential function is substituted into this ODE. The resultant polynomial equation is solved by using algebraic operations. The method works for the Jimbo-Miwa, the Zakharov-Kuznetsov and the modified Zakharov-Kuznetsov equations in conformable time fractional forms. All the solutions are expressed in explicit forms.

P-band radar remote sensing applied during the Airborne Microwave Observatory of Subcanopy and Subsurface (AirMOSS) mission has shown great potential for estimation of root zone soil moisture. When retrieving the soil moisture profile (SMP) from P-band radar, a mathematical function describing the vertical moisture distribution is required. Because only a limited number of observations are available, the number of free parameters of the mathematical model must not exceed the number of observed data. For example, a second order polynomial that contains 3 free parameters was presumed based on in-situ SMP data. The polynomial is currently parameterized based on 3 backscatter observations provided by AirMOSS (i.e. one frequency at three polarizations of HH, VV and HV). In this paper, a more realistic, physically-based SMP model containing 3 free parameters is derived based on a solution to Richards’ equation for unsaturated flow in soils. Evaluation of the new SMP model based on both numerical simulations and measured data revealed that it exhibits greater flexibility for fitting measured and simulated SMPs than the currently applied polynomial. It is also demonstrated that the new SMP model can be reduced to a second order polynomial at the expense of fitting accuracy.

The exact solutions of some conformable time fractional PDEs are presented explicitly. The modified Kudryashov method is applied to construct the solutions to the conformable time fractional Regularized Long Wave-Burgers (RLW-Burgers, potential Korteweg-de Vries (KdV) and clannish random walker's parabolic (CRWP) equations. Initially, the predicted solution in the finite series of a rational form of an exponential function is substituted to the ODE generated from the conformable time fractional PDE by using wave transformation. The coefficients used in the finite series are determined by solving the algebraic system derived from the coefficients of the powers of the predicted solution.

Recently, there has been a growing interest in the production of electricity from renewable energy sources (RES). The RES investment is characterized by uncertainty, which is long-term, costly, depend on feed-in-tariff and support schemes. In this paper, we address the real option valuation (ROV) of a solar power plant investment. The real option framework is investigated. This framework considers the renewable certificate price, furthermore the cost of delay between establishing and operating the solar power plant. The optimal time of launching the project and assess the value of deferred option are discussed. The new three stage numerical methods are constructed, the Lobatto3C-Milstein (L3CM) methods. The numerical methods are integrated with concept of Black-Scholes option pricing theory, and applied in option valuation for solar energy investment with uncertainty. The numerical results of L3CM, finite difference and Monte Carlo methods are compared to show the efficiency of our methods. Our data set refers to the Arab Republic of Egypt.

A Natural gas is increasingly being sought after as a vital source of energy, given that its production is very cheap and does not cause the same environmental harms that other resources, such as coal combustion, do. Understanding and characterizing the behavior of natural gas is essential in hydrocarbon reservoir engineering, natural gas transport, and process. Natural gas compressibility factor, as a critical parameter, defines the compression and expansion characteristics of natural gas under different conditions. In this study, a simple second-order polynomial model based on the group method of data handling (GMDH) is presented to determine the compressibility factor of different natural gases at different conditions, using corresponding state principles. The accuracy of the model evaluated through graphical and statistical analyses. The results show that the model is capable of predicting natural gas compressibility with an average absolute error of only 2.88%, a root means square of 0.03, and a regression coefficient of 0.92. The performance of the developed model compared to widely known, previously published equations of state (EOSs) and correlations, and the precision of the results demonstrates its superiority over all other correlations and EOSs.

The three dimensional conformable time fractional Kadomtsev-Petviashvili and the conformable time fractional modified Kawahara equations are solved by implementing the Kudryashov's procedure. The corresponding wave transformation reduces both equations to some ODEs. Balancing the nonlinear and the highest order derivative terms gives the structure of the solutions in the finite series form. The useful symbolic tools are used to solve the resultant algebraic systems. The solutions are expressed in explicit forms.

In this paper we use the conformable fractional derivative to discuss some fractional linear differential equations with constant coefficients. By applying some similar arguments to the theory of ordinary differential equations, we establish a sufficient condition to guarantee the reliability of solving constant coefficient fractional differential equations by the conformable Laplace transform method. Finally, we analyze the analytical solution for a class of fractional models associated with Logistic model, Von Foerster model and Bertalanffy model is presented graphically for various fractional orders and solution of corresponding classical model is recovered as a particular case.

In the paper, the authors view some ordinary differential equations and their solutions from the angle of (the generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius-Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds.

The exact solutions in the wave form are derived for the time fractional KdV and the time fractional Burgers' equations in conformable fractional derivative sense. The fractional variable change using the fundamental properties of the conformable derivative reduces both equations to some nonlinear ODEs. The predicted solution is assumed to be a finite series form of a function satisfying a particular first-order ODE whose solution contains an exponential function in the denominator. The solutions are represented in the explicit forms and illustrated by some choices of the parameters for various fractional orders of the equations.

In the note, the authors find several simple and explicit forms for a family of inhomogeneous linear ordinary differential equations studied in "D. Lim, Differential equations for Daehee polynomials and their applications, J. Nonlinear Sci. Appl. 10 (2017), no. 4, 1303-1315; Available online at http://dx.doi.org/10.22436/jnsa.010.04.02".

Blood perfusion quantification is important vital parameters in different diagnostic procedure, using infrared thermography imaging; it is reliable to use this technique as non-contact, non-invasive blood flow measurement method. Therefore, we developed a measurement protocol for blood flow over the arm's anterior surface. By using the superficial brachial and radial veins to be monitored under the impact of cold-excitation of (2 °C to 5 °C), the blood perfusion signal was detected using thermal imager of long-wave infrared spectral range (LWIR, 7μm - 14 μm). The simulation of Penne's bioheat transfer equation was performed to be compared with results obtained from the infrared thermography. Furthermore, the proposed blood flow monitoring using external adjusting of the excitation temperature, by using (cold-compress, or cold air-stream) applied to the region under testing. The signal detected resembles to the hemodynamic pulse of the superficial veins, in the definition of systolic and diastolic phases of the cardiac cycle. Moreover, statistical analysis applied to the BFIRT signals from 24 subjects to estimate the skin's mean temperature after recovery from the thermal excitation.

There are numerous approaches to model an influence of manufacturing parameters on the dissolution and disintegration of solid dosage forms. A modern approach for modelling and simulating complex and heterogeneous systems such as pharmaceutical tablets focuses on computationally-intensive finite-element or discreet element methods. Industrial challenges such as Process Analytical Technology (PAT) and Quality by Design are stimulating computer-based technologies to design, control and improve the quality of pharmaceutical compacts and their performance. In this study the disintegration and dissolution model based on the numerical solutions of the Noyes-Whitney equation and cellular automata supported disintegration model are proposed and evaluated. The results from in vitro release studies of the mefenamic acid formulations were compared to the calculated release patterns from the tablet structures obtained from microtomography experiments and the purely algorithmically created virtual tablets.

This paper describes the development of a two-dimensional water quality model that solves hydrodynamic equations tied to transport equations with reactions mechanisms inherent in the processes. This enable us to perform an accurate assessment of the pollution in a coastal ecosystem. The model was developed with data drawn from the ecosystem found in Mexico's southeast state of Tabasco. The coastal ecosystem consists of the interaction of El Yucateco lagoon with the Chicozapote and Tonalá rivers, that connect the lagoon with the Gulf of Mexico. We present the results of pollutants transport simulation in the coastal ecosystem, focusing on toxic parameters for two hydrodynamic scenarios: wet and dry seasons. As it of interest in the zone, we study the transport of four metals: Cadmium, Chromium, Nickel and Lead. In order to address our objectives we solved numerically a self-posed mathematical problem,which is based on the measured data. The performed simulations show to characterise metal transport within the acceptable range of accuracy and in accordance with the measured data. The performed simulations show to characterise metals transport with an acceptable accuracy, agreeing well with measured data in total concentrations in four control points along the water body. Although for the accurate implementation of the hydrodynamic-based water quality model herein presented, boundary (geometry, tides, wind, etc.) and initial (concentrations measurements) conditions are required, it poses as an excellent option when the distribution of solutes with high accuracy is required, easing environmental, economic and social management of coastal ecosystems.

In this paper, we consider a fourth-order suspension bridge equation with nonlinear damping term |u_t|^m-2u_t and source term |u|^p-2u. We give necessary and sufficient condition for global existence and energy decay results without considering the relation between m and p. Moreover, when p>m, we give sufficient condition for finite time blow-up of solutions. The lower bound of the blow-up time T_max is also established. It worth to mention that our obtained results extend the recent results of Wang (J. Math. Anal. Appl., 2014) to the nonlinear damping case.

Inverse Sumudu transform multiple shifting properties are used to design methodology for solving ordinary differential equations. Then algorithm applied to solve Whittaker and Zettl equations to get their new exact solutions and profiles which shown through Maple complex graphicals. Table of inverse Sumudu transforms for elementary functions given for supporting the differential equations solving using inverse Sumudu transform.

Today, Colebrook’s equation is mostly accepted as an informal standard for modeling of turbulent flow in hydraulically smooth and rough pipes including transient zone in between. The empirical Colebrook’s equation relates the unknown flow friction factor (λ) with the known Reynolds number (R) and the known relative roughness of inner pipe surface (ε/D). It is implicit in unknown friction factor (λ). Implicit Colebrook’s equation cannot be rearranged to derive friction factor (λ) directly and therefore it can be solved only iteratively [λ=f(λ, R, ε/D)] or using its explicit approximations [λ≈f(R, ε/D)]. Of course, approximations carry in certain error compared with the iterative solution where the highest level of accuracy can be reached after enough number of iterations. The explicit approximations give a relatively good prediction of the friction factor (λ) and can reproduce accurately Colebrook’s equation and its Moody’s plot. Usually, more complex models of approximations are more accurate and vice versa. In this paper, numerical values of parameters in various existing approximations are changed (optimized) using genetic algorithms to reduce maximal relative error. After this improvement computational burden stays unchanged while accuracy of approximations increases in some of the cases very significantly.

By using Modified simple equation method, we study the Cahn Allen equation which arises in many scientific applications such as mathematical biology, quantum mechanics and plasma physics. As a result, the existence of solitary wave solutions of the Cahn Allen equation is obtained. Exact explicit solutions interms of hyperbolic solutions of the associated Cahn Allen equation are characterized with some free parameters. Finally, the variety of structure and graphical representation make the dynamics of the equations visible and provides the mathematical foundation in mathematical biology, quantum mechanics and plasma physics.

Allometric equations are used to estimate accurate biomass and carbon stock of forests. However, in Ethiopia only few allometric equations as compared to its floral diversity and species-specific allometric equations for Acacia species are still not developed in Ethiopia. The numbers of tree marked for sampling are Fifty-four (54) using preferential sampling. Diameter at breast height, wood density and tree height were collected as independent variables to predict species specific dry biomass of Acacia species. The new species-specific allometric models have been performed using linear regression analysis in the R software. The Above ground biomass (AGB) have been validated using quantitative statically using the pantropic model. Six candidate models have been developed for each species and four best models for each species of dry biomass was selected based on goodness-of-fit statistics and equation performance analysis of the candidate models. The best model for predicting above ground biomass for Acacia seyal is 0.20636*((DBH²)Hρ) ^0.53167, for Acacia polyacantha is 7.26982((DBH)²Hρ)^0.21750, for Acacia ethibcia is 29.01898*((DBH)²Hρ)^0.21518 and for Acacia toritolis is 3.82427*((DBH)²Hρ)^0.16748. The selected models are the best performing (P> 0.01) and higher adjusted R² (>80%) and has lower Akaike’s Information Criteria (AIC) and residual standard error (RSE) values as comparing the rest of the model. The validation of new developed biomass model using Tukey test indicated that significant variation of mean biomass (P<0.05) between the new developed model and the generalized model. The statistics model performance analysis of Nash-Sutcliffe efficiency (NSE) value is approaching to one, indicating that the new developed model has better performance model as compared with generalized model. Moreover, the percent bias of the new developed models is close to zero which indicates that the site-specific biomass models have more accurate estimator and the generalized biomass models have overestimated biomass for the four Acacia species.

The Sine-Gordon expansion method is implemented to construct exact solutions some conformable time fractional equations in Regularized Long Wave(RLW)-class. Compatible wave transform reduces the governing equation to classical ordinary differential equation. The homogeneous balance procedure gives the order of the predicted polynomial-type solution that is inspired from well-known Sine-Gordon equation. The substitution of this solution follows the previous step. Equating the coefficients of the powers of predicted solution leads a system of algebraic equations. The solution of resultant system for coefficients gives the necessary relations among the parameters and the coefficients to be able construct the solutions. Some solutions are simulated for some particular choices of parameters.

Biohydrogen is a versatile energy carrier for the generation of electric energy from renewable sources. Hydrogenases can be used in enzymatic fuel cells to oxidize dihydrogen. The rate of electron transfer (ET) at the anodic side between the [NiFe]-hydrogenase enzyme distal iron-sulfur cluster and the electrode surface can be described by the Marcus equation. All parameters for the Marcus equation are accessible from DFT calculations. The distal cubane FeS-cluster has a three cysteine and one histidine coordination [Fe₄S₄](His)(Cys)₃ first ligation sphere. The reorganization energy (inner- and outer-sphere) is almost unchanged upon a histidine-to-cysteine substitution. Differences in rates of electron transfer between the wild-type enzyme and the all-cysteine mutant can be rationalized by a diminished electronic coupling between the donor and acceptor molecules in the [Fe₄S₄](Cys)₄ case. The fast and efficient electron transfer from the distal iron-sulfur cluster is realized by a fine-tuned protein environment which facilitates the flow of electrons. This study enables the design and control of electron transfer rates and pathways by protein engineering.

In the paper, the authors present an explicit form for a family of inhomogeneous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomogeneous linear ordinary differential equations in terms of the Lerch transcendent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomogeneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function.

In the paper, by virtue of techniques in combinatorial analysis, the authors simplify two families of nonlinear ordinary differential equations in terms of the Stirling numbers of the first kind.

The goal of the study was to assess the farmers’ acceptance of three sustainable production strategies, namely ‘Agro-forestry’, ‘Alternative protein source’ and ‘Prolonged maternal feeding’. Data on the acceptance of these strategies were collected by a survey of dairy farmers in six EU countries (AT, BE, DK, FI, IT, UK). An extended version of the Technology Acceptance Model (TAM) was applied by means of Structural Equation Modelling to testing various hypotheses on attitudes and intentions of dairy farmers towards these novel production strategies, as well as the influence of organic practices and collaborative behaviours along the supply chain. We found that the most preferred strategy - across all countries - was soy substitution by alternative protein sources. We also found that the intention to adopt a sustainable production strategy may derive from the influence of opinions (and behaviours) of relevant others, showing the role of interactions among farmers and other stakeholders in the adoption of innovations. Finally, the perceived usefulness of all investigated strategies is higher for organic farmers, while collaborative patterns reduce the impact of subjective norm on usefulness and overall acceptance. Our findings should encourage policy makers to consider the important role of supply chain management practices, including collaboration, to enhance the sustainability of dairy farming systems.

Social media and other web 2.0 tools have provided users the platform to interact and also disclose personal information not only with their friends and acquaintances, but also with relative strangers with unprecedented ease. This has enhanced the ability of people to share more about themselves, their families, and their friends through a variety of media including text, photo, and video, thus developing and sustaining social and business relationships. The purpose of the paper is to identify the factors that predict self-disclosure on social networking sites within the Ghanaian context. Data was collected from 452 students in three leading universities in Ghana and analyzed with Partial Least Square-Structural Equation Modeling. Results from the study revealed that all variables in the proposed model with the exception of interaction and perceived control were significant predictors of self-disclosure with privacy risk being the most significant predictor. In all, the model accounted for 54.6 percent of the variance in self disclosure. The implications and limitations of the current study are discussed and directions for future research proposed.

The material derivative is important in continuum physics. This Letter shows that the expression $\frac{d }{dt}=\frac{\partial }{\partial t}+(\bm v\cdot \bm \nabla)$, used in most literature and textbooks, is incorrect. The correct expression $ \frac{d (:)}{dt}=\frac{\partial }{\partial t}(:)+\bm v\cdot [\bm \nabla (:)]$ is formulated. The solution existence condition of Navier-Stokes equation has been proposed from its form-solution, the conclusion is that "\emph{The Navier-Stokes equation has a solution if and only if the determinant of flow velocity gradient is not zero, namely $\det (\bm \nabla \bm v)\neq 0$.}"

The main obstacle affecting the natural and herbal products industry is the infringement of trade patents. This resulted in losing the opportunity to compete with others in this business. Creating additional value to this business is, therefore, a way of leading to patenting for trade protection for entrepreneurs. This study was conducted to develop the structural equation modeling of guidelines for creating additional value for Thai herbs to support trade patent. The study comprises a mixed method with three parts. In the first part, data comes from interviews with selected 9 specialists. The second part was supported by interviews to create the questionnaire and distributed it to 500 entrepreneurs, the next step was the data analysis using structural equation modeling (SEM). The third part used data from the SEM model to discussion focused on developing the conceptual model by 7 experts. The analysis of the developed model passed the criteria for evaluating consistency and empirical data. All hypotheses proposed in this study were supported, the findings revealed that all factors had a direct effect at a statistically significant level of 0.001. Furthermore, this investigation allows knowing the Brand Identity factor were practically important for creating value of Thai herbs leading to patent protection.

Artificial intelligence (AI) applications in different fields are developing rapidly, among which AI painting technology, as an emerging technology, has received wide attention from users for its creativity and efficiency. This study aimed to investigate the factors that influence user acceptance of the use of AIBPS by proposing an extended model that combines the Extended Technology Acceptance Model (ETAM) with the AI-based Painting System (AIBPS).A questionnaire was administered to 528 Chinese participants, using validated factor analysis data and Structural Equation Modeling (SEM) was used to test the hypotheses. The findings showed that hedonic motivation (HM) and perceived trust (PE) had a positive effect (+) on users' perceived usefulness (PU) and perceived ease of use (PEOU), while previous experience (PE) and technical features (TF) had no effect (-) on users' perceived usefulness (PU). This study provides an important contribution to the literature on AIBPS and the evaluation of systems of the same type, which helps to promote the sustainable development of AI in different domains and provides a possible space for further extension of TAM, thus helping to improve the user experience of AIBPS. The results of the study provide insights for system developers and enterprises to better motivate users to use AIBPS.

This work focuses on the bifurcation behavior before chaos phenomenon happens. Traditional numerical method is unable to solve the unstable limit cycle of nonlinear system. One algorithm is introduced to solve the unstable one, which is based one of the continuation method is called DEPAR approach. Combined with analytic and numerical method, the two stable and symmetrical equilibrium solutions exist through Fork bifurcation and the unstable and symmetrical limit cycles exist through Hopf bifurcation of Lorenz system. With the continuation algorithm, the bifurcation behavior and its phase diagram is solved. The results demonstrate the unstable periodical solution is around the equilibrium solution, besides the trajectory into the unstable area cannot escape but only converge to the equilibrium solution.

GNSS have been widely used in navigation, positioning and timing. Nowadays, the multipath errors previously considered detrimental may be re-utilized for the remote sensing of geophysical parameters (soil moisture, vegetation and snow depth), e.g. GPS- Multipath Reflectometry (GPS-MR). In this paper, a new element describing bistatic scattering properties of vegetation is incorporated into the traditional GPS-MR model. This new element is the first-order radiative transfer equation model. The new forward GPS multipath simulator is able to explicitly link the vegetation parameters with GPS multipath observables (signal-to-noise-ratio (SNR), code pseudorange and carrier phase observables). The trunk layer and its corresponding scattering mechanisms are ignored since GPS-MR is not suitable for high forest monitoring due to the coherence of direct and reflected signals. Based on this new model linking the GPS observables (SNR, phase and pseudorange) with detailed vegetation parameters, the developed simulator can present how the GPS signals (L1 and L2 carrier frequencies, C/A, P(Y) and L2C modulations) are transmitted (scattered and absorbed) through vegetation medium and received by GPS receivers. Simulation results show that wheat will decrease the amplitudes of GPS multipath observables, if we increase the vegetation moisture contents or the scatters sizes (stem or leaf), the amplitudes of GPS multipath observables (SNR, phase and code) decrease. Although the Specular-Ground component dominates the total specular scattering, vegetation covered ground soil moisture has almost no effects on the final multipath signatures. Our simulated results are consistent with published results for environmental parameter detections with GPS-MR.

Inherent in every Active Magnetic Bearing (AMB) are differences between the expected geometric axes and the actual magnetic axes due to a combination of discrepancies, including physical variation from manufacturing tolerances and misalignment from mechanical assembly, fringing and leakage effects, as well as variations in magnetic material properties within a single AMB. A method is presented here for locating the magnetic axes of an AMB that will facilitate the accurate characterization of the bearing air gaps for potential improvement in field tuning, performance analyses and certain shaft force measurement techniques. This paper presents an extension of the application of the bias current perturbation method for the determination of the magnetic center [4] to the determination of magnetic axes for the further development of accurate current-based force measurement techniques [1].

Since crystals are made of periodic structures in space, predicting their three period vectors starting from any values based on the inside interactions is a basic theoretical physics problem. For the general situation where crystals are under constant external stress, we derived dynamical equations of the period vectors in the framework of Newtonian dynamics, for pair potentials recently (doi:/10.1139/cjp-2014-0518). The derived dynamical equations show that the period vectors are driven by the imbalance between the internal and external stresses. This presents a physical process where when the external stress changes, the crystal structure changes accordingly, since the original internal stress can not balance the external stress. The internal stress has both a full kinetic energy term and a full interaction term. It is extended to many-body interactions in this paper. As a result, all conclusions in the pair-potential case also apply for many-body potentials.

The paper is considering of the basic differential equations of hydrodynamics: the equation of continuity and motion. On the simplest example, it is shown that the equation of continuity in a system with the equation of motion leads to contradictions and erroneous results of modeling. A more correct form of the continuity equation is described. It is shown that the equations of motion can be written in the form of complete differentials. Three possible integral forms of the equations of motion are presented. As a conclusion, the existence and smoothness of the solution of the Navier-Stokes equations are considered.

In the paper, by virtue of the Faà di Bruno formula, some properties of the Bell polynomials of the second kind, and an inversion formula for the Stirling numbers of the first and second kinds, the authors establish meaningfully and significantly two identities which simplify coefficients in a family of ordinary differential equations associated with higher order Bernoulli numbers of the second kind.

In the paper, by the Faά di Bruno formula, several identities for the Bell polynomials of the second kind, and an inversion theorem, the authors simplify coefficients of two families of nonlinear ordinary differential equations for the generating function of the Catalan numbers and discover inverses of fifteen closely related lower triangular integer matrices.

We extended the usage of the expansion method based on Sine-Gordon equation to the two dimensional Fisher equation. The relation between the trigonometric and hyperbolic functions are derived from the Sine-Gordon equation dened in two space dimension. The complex-valued traveling wave solutions to the two dimensional Fisher and Nagumo equations are set in forms a nite series of multiplications of powers of sech(.) and tanh(.) functions.

Restoration experience is a positive response against the ever-increasing problem of stress. It is recovery of attentional capacity, clearing random thoughts and feelings of relaxation and calmness. Although there is a body of research on investigating the impact of open green space characteristics on psycho- logical restoration through attention restoration theory and supportive environment theory, there is little knowledge on how the interaction among these characteristics could promote restorative outcomes. To address this problem, using Perceived Sensory Dimension (PSD) for open green space qualities and Perceived Restorativeness (PR) as the property of restorative environment, this study assessed the extent to which the PSD and PR impact on restoration experience. Using 444 screened surveys, a Partial Least Square Structural Equation Model (PLS-SEM) was developed and the validity and reliability of it, was demonstrated. Extensive analysis of the results showed how environmental qualities can promote restoration experience using a sample of university students. These results provide information for landscape architecture and planning researchers to promote the development of open spaces as a resource of psychological restoration and stress relief.

The ordinary continuity equation relating the current and density of a system is extended to incorporate systems with dual (longitudinal and transverse) currents. Such a system of equations is found to have the same mathematical structure as that of Maxwell equations. The horizontal and transverse currents and the densities associated with them are found to be coupled to each other. Each of these quantities are found to obey a wave equation traveling at the velocity of light in vacuum. London's equations of super-conductivity are shown to emerge from some sort of continuity equations. The new London's equations are symmetric and are shown to be dual to each other. It is shown that London's equations are Maxwell's equations with massive electromagnetic field (photon). These equations preserve the gauge invariance that is broken in other massive electrodynamics. The duality invariance may allow magnetic monopoles to be present inside superconductors. The new duality is called the comprehensive duality transformation.

Proposed in 1889 by Svante Arrhenius, the Arrhenius equation is an important result in physical chemistry which aims to find the relationship between temperature and reaction rates. It is also an essential equation for analytic chemistry which helps to determine the energy barrier for a given reaction. In this article, we will go through a detailed derivation of the exponential factor in the Arrhenius Equation based on the Boltzmann distribution of particle energy probability. Then, we will attempt to model the pre-exponential factor in accordance with the principles of collision theory for bi-molecular reaction with gaseous reactants. Understanding the basic principles of chemical reaction provides valuable tools to analyze complex reaction mechanisms in the perspective of optimizing existing industrial processes.

The space time fractional Korteweg-de Vries equation, modified Equal Width equation and Benney-Luke Equations are solved by using simple hyperbolic tangent Ansatz method. A simple compatible wave transformation in one dimension is employed to reduce the governing equations to integer ordered ODEs. Then, the ansatz approximation is used to derive exact solutions. Some illustrative examples are presented for some particular choices of parameters and derivative orders.

In the paper, by virtue of the Fa`a di Bruno formula, some properties of the Bell polynomials of the second kind, and the inversion formulas of binomial numbers and the Stirling numbers of the first and second kinds, the authors simplify meaningfully and significantly coefficients in two families of ordinary differential equations associated with higher order Frobenius–Euler numbers.

The potential of Thai industrial product design is still inferior to those of leading competitors in world market that give more importance on the design during their product development to increase their competitive edges on commercial scale. The product design is very important part for sustainable growth in this industry. Thus, this research aims at investigating footwear design strategies for Thai footwear industry to be excellence in world market. The research has been designed with the mixed method of both qualitative and quantitative study. The quantitative data were collected through semi-structure interview from 500 designers who presented their designs to join the award competition. The results revealed that the footwear design strategies consisted of 4 factors, i.e. 1) design, 2) market analysis, 3) innovation, and 4) information technology. This paper utilizes the method of Structural Equation Modeling (SEM) to establish a strategies model for competitive advantage in Thai footwear industry. The analysis results indicated that the footwear design strategies model could help make more effective policies and organization strategies for enterprises and designers to develop themselves to be excellence in world market.

In this paper, some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral is used to get two new representations of the fundamental solution in form of the Mellin convolution of the special functions of the Wright type. Moreover, some new closed form formulas for particular cases of the fundamental solution are derived. In particular, we solve an open problem of representation of the fundamental solution to the two-dimensional neutral-fractional diffusion-wave equation in terms of the known special functions.

Complex and real valued exact solutions to some reaction-diffusion equations are suggested by using homogeneous balance and Sine-Gordon equation expansion method. The predicted solution of finite series of some hyperbolic functions are determined by using some relations between the hyperbolic functions and trigonometric functions based on Sine-Gordon equation and traveling wave transform. The Newel-Whitehead-Segel and Zeldovich equations are solved explicitly. Some real valued solutions are depicted for some particular choice of parameters.

In the present work the problem of the attenuation of longitudinal sound oscillations in a conducting medium are considered. The proposed approach is based on the dynamic interaction of electron gas with the lattice vibrations. This interaction is manifested in the modification of kinetic equation for electrons. The process is accompanied by generation of an electric field.

In this paper we study the initial boundary value problem for logarithmic Higher Order Wave equation. Introducing the Logarithmic Sobolev inequality and using the combination of Galerkin method, we consider the theorem of existence of a global weak solution to problem for the initial boundary value problem of the logarithmic wave equation. By constructing an appropriate Lyapunov function, we obtain the decay estimates of energy for logarithmic Higher Order Wave equation. The proof of the main theorem is given.

As a result of rapid population growth, an exponentially growing human population, and industrial expansion, it has become increasingly difficult to manage municipal solid wastes throughout the world. Decentralized waste management systems have created difficult situations in developing countries such as Malaysia. Wastes generated in the country, due to various cultural, social, and religious activities, organic and contributing to environmental pollution (air, water, and soil) and human health troubles. A questionnaire survey was participated by 220 construction professionals in Malaysia using structured and semi-structured methods. The framework was assessed using A partial least square structural equation modeling (PLS-SEM) to target sustainable development goals (SDG). Statistical analysis results indicate a significant effect between SCW management, since(r(270)=.687, P&lt;0.001). Improving factors has strong relationship with SCW management, since(r(270)=.723, P&lt;0.001). The mediation results also suggested a significant indirect positive effect of improving factors drivers on SCW management through policy-related factors sinceβ=0.688, t=8.254, P&lt;0.001, 95% CI for β=0.536,0.866. Finally, policy-related factors construct has a strong relationship with SCWM) management, since(r(270)=.811, P&lt;0.001) With the R Square of 0.787 and 0.785. The developed framework can improve construction waste management in the construction industry and enhance construction waste management to achieve global sustainable development goals. The findings show that one of the most critical issues of enhancing profitability is using preventive policies to reduce construction waste. This study could guide construction industry stakeholders in identifying the different waste management features during a building project's construction and design stage

Today, two very efficient methods for calculation of flow distribution per branches of a looped gas pipeline are available. Most common is improved Hardy Cross method, while the second one is so-called unified node-loop method. For gas pipeline, gas flow rate through a pipe can be determined using Colebrook equation modified by AGA (American Gas Association) for calculation of friction factor accompanied with Darcy-Weisbach equation for pressure drop and second approach is using Renouard equation adopted for gas pipeline calculation. For the development of Renouard equation for gas pipelines some additional thermodynamic properties are involved in comparisons with Colebrook and Darcy-Weisbach model. These differences will be explained. Both equations, the Colebrook’s (accompanied with Darcy-Weisbach scheme) and Renouard’s will be used for calculation of flow through the pipes of one gas pipeline with eight closed loops which are formed by pipes. Consequently four different cases will be examined because the network is calculated using improved Hardy Cross method and unified node-loop method. Some remarks on optimization in this area of engineering also will be mentioned.

Using a nonclassical symmetry of nonlinear reaction-diffusion equations, some exact multi-dimensional time-dependent solutions are constructed for a fourth-order Allen-Cahn-Hilliard equation. This models a phase field that gives a phenomenological description of a two-phase system near the critical temperature. Solutions are given for the changing phase of a cylindrical or spherical inclusion, allowing for a 'mushy zone' with mixed state that is controlled by imposing a pure state at the boundary. The diffusion coefficients for transport of one phase through the mixture, depend on the phase field value, since the physical structure of the mixture depends on the relative proportions of the two phases. A source term promotes stability of both of the pure phases but this tendency may be controlled or even reversed through the boundary conditions.

In this paper, we work with a diffusion-perturbed risk model comprising a surplus generating process and an investment return process. The investment return process is of standard Black-Scholes type, that is, it comprises a single risk-free asset that earns interest at a constant rate and a single risky asset whose price process is modelled by a geometric Brownian motion. Additionally, the company is allowed to purchase noncheap proportional reinsurance priced via the expected value principle. Using the Hamilton-Jacobi-Bellman approach, we derive a second-order Volterra integrodifferential equation which we transform into a linear Volterra integral equation of the second kind. We proceed to solve this integral equation numerically using the block-by-block method for the optimal reinsurance retention level that minimizes the ultimate ruin probability. The numerical results based on light- and heavy-tailed distributions show that proportional reinsurance and investments play a vital role in enhancing the survival of insurance companies. But the ruin probability exhibits sensitivity to the volatility of the stock price.

Plants have evolved tightly regulated strategies to adapt and acclimate to a changing environment to ensure their survival. Various environmental factors affect plant distribution, growth and yield. Low temperature belongs to those abiotic factors which significantly constrain range boundaries of plant species. Exposing plants to low but non-freezing temperature induces a multigenic processes termed cold acclimation, which finally results in an increased freezing tolerance. Cold acclimation comprises reprogramming of the transcriptome, proteome and metabolome and affects communication and signaling between subcellular organelles. Reprogramming of the central carbohydrate metabolism plays a key role in cold acclimation. This review summarizes current knowledge about the role of carbohydrate metabolism in plant cold acclimation. A focus is laid on subcellular metabolic reprogramming, its thermodynamic constraints under low temperature and mathematical modelling of metabolism.

New exact solutions to conformable time fractional EW and modified EW equations are constructed by using Sine-Gordon expansion approach. The fractional traveling wave transform and homogeneous balance have significant roles in the solution procedure. The predicted solution is of the form of some finite series of multiplication of powers of cos and sin functions. The relation among trigonometric and hyperbolic functions in sense of Sine-Gordon expansion gives opportunity to construct the solutions in terms of hyperbolic functions.

A probabilistic description is essential for understanding the dynamics of stochastic systems far from equilibrium, given uncertainty inherent in the systems. To compare different Probability Density Functions (PDFs), it is extremely useful to quantify the difference among different PDFs by assigning an appropriate metric to probability such that the distance increases with the difference between the two PDFs. This metric structure then provides a key link between stochastic systems and information geometry. For a non-equilibrium process, we define an infinitesimal distance at any time by comparing two PDFs at times infinitesimally apart and sum these distances in time. The total distance along the trajectory of the system quantifies the total number of different states that the system undergoes in time, and is called the information length. By using this concept, we investigate the information geometry of non-equilibrium processes involved in disorder-order transitions between the critical and subcritical states in a bistable system. Specifically, we compute time-dependent PDFs, information length, the rate of change in information length, entropy change and Fisher information in disorder-to-order and order-to-disorder transitions, and discuss similarities and disparities between the two transitions. In particular, we show that the total information length in order-to-disorder transition is much larger than that in disorder-to-order transition, and elucidate the link to the drastically different evolution of entropy in both transitions. We also provide the comparison of the results with those in the case of the transition between the subcritical and supercritical states and discuss implications for fitness.

Recently, the awareness of energy conservation, carbon emission reduction, and environmental protection in Taiwan has begun to gradually improve, and bicycles have become an important vehicle for residents. In 2009, Kaohsiung began promoting public bicycle sharing. The number of users in that year amounted to 140,000. By 2015, the accumulated number of users reached nine million. This study adopted the Theory of Planned Behavior (TPB) to explore the factors influencing the behavioral intentions of public bicycle users, including the three dimensions of attitude, subjective norm, and perceived behavioral control, and probed into the relationship between the factors influencing the behavioral intentions of public bicycle users in Kaohsiung City and the dimensions. This study used a questionnaire to collect data and SPSS & AMOS for statistical analysis. The empirical results demonstrated that the most prominent factors influencing the behavioral intentions of public bicycle users in Kaohsiung City included perceived behavioral control, attitude, and subjective norm, in order. Among the influencing factors in the three dimensions, personal interests, main groups, and self-competence had a positive and significant correlation. Administrators should provide more incentives related to the three influencing factors to encourage the public in Kaohsiung City to use public bicycles more frequently as their short-distance travel mode, so as to achieve the purposes of saving energy, reducing carbon emissions, and protecting the urban environment.

We apply principles of Gibbs phase plane chemistry to and across the entire ocean-atmospheric interface. Surface tension increments support a two dimensional, tangential pressure well known to determine rates of bulk gas, bubble, salt, spray and momentum transfer plus both sensible and latent heat fluxes. Hence it is worth asking whether tension mapping follows from current understanding of two dimensional composition. A history is provided dating back centuries and demonstrating that detrital organic macromolecules are central; subtle surfactant functional variation creates a microforcing field which dissipates turbulent energy at the sub-meter scale. Since we have just distributed major biopolymeric classes emitted as primary organic aerosol, further climate links can be established by considering full planar thermochemistry. Organic microlayer behaviors are reviewed with attention to confined, analog phase transitions among two dimensional “solid, liquid, (and) gaseous” states serving as elasticity indicators. We also discuss surfactant properties of general marine dissolved organic carbon, demonstrating that only proteins and lipids are capable of occupying significant local micro-area. The literature often suggests albumin and stearic acid as best proxies, and so we distribute their concentrations through multilevel global ecodynamic simulations. Consensus distributions are obtained in order to control adsorptive equilibria. Working from conservation of planar free energy, a parametric equation of state is devised relating excess coverage to the surface pressure-modulus. Constant settings for the proxy pair are drawn from laboratory study, and they successfully reproduce frequencies for surfactant solid-to-gas occurrence in ambient compression experiments. Functionally resolved organic measurements are rare and so we group them into super-ecological province tables showing that our bulk concentration estimates are reasonable. Outputs are then fed into a coverage-tension-elasticity code. Resulting contours traverse the critical range for piston velocity, bubble-spray and damping effects on either a regional or seasonal basis. There is also a possibility for widespread microlayer crystallization in polar seas. The concepts are a direct extension of our organic aerosol work, and the two approaches could be inserted into Earth System Models in tandem. Uncertainties in the logic are enumerated and include kinetic and thermochemical factors at multiple scales. But the problems are reducible through molecular modeling coupled to renewed laboratory and field study. Connections to marine colloids-gels, microlayer iron chelation, and linings of the ice channel network are discussed additionally.

An inverse aeroacoustic problem for a helicopter rotor combined with aerodynamic constraint is proposed based on Ffowcs Williams and Hawkings equation in subsonic. The rotor noise includes thickness noise and loading noise when quadrupole noise is neglected. Thickness noise is related to geometry and motion conditions. Loading noise is related to the pressure on the wall. Therefore, the equation between pressure on the wall and far-field noise can be established, thus the pressure on the wall can be obtained by solving this equation. Since this equation is an ill-posed, the singular value decomposition combined with the regulation method is applied and the aerodynamic constraint is taken into account. The direct noise prediction is verify firstly and then the inverse problem is solved. The reconstruction pressure is compared to the input data. The result is in good agreement with the input value. At the same time, the influence of interference noise is also considered. Under low signal-to-noise ratio, the reconstruction result is also reasonable.

In this paper only one basic assumption has been made: if we try to describe black holes, their behavior should be understood in the same language as the one we use for particles; black holes should be treated just like atoms. They must be quantum forms of matter, moving in accordance with Schrödinger equations just like everything else. In particular, Rosen’s quantization approach to the gravitational collapse is applied in the simple case of a pressureless “star of dust” by finding the gravitational potential, the Schrödinger equation and the solution for the collapse’s energy levels. By applying the constraints for a Schwarzschild black hole (BH) and by using the concept of BH effective state, previously introduced by one of the authors (CC), the BH quantum gravitational potential, Schrödinger equation and the BH energy spectrum are found. Remarkably, such an energy spectrum is in agreement (in its absolute value) with the one which was conjectured by Bekenstein in 1974 and consistent with other ones in the literature. This approach also permits to find an interesting quantum representation of the Schwarzschild BH ground state at the Planck scale. Moreover, two fundamental issues about black hole quantum physics are addressed by this model: the area quantization and the singularity resolution. As regards the former, a result similar to the one obtained by Bekenstein, but with a different coefficient, has been found. About the latter, it is shown that the traditional classical singularity in the core of the Schwarzschild BH is replaced, in a full quantum treatment, by a two-particle system where the two components strongly interact with each other via a quantum gravitational potential. The two-particle system seems to be non-singular from the quantum point of view and is analogous to the hydrogen atom because it consists of a “nucleus” and an “electron”.

Bringing toghether mathematical and philosophical ideas related to transcendental numbers, this paper begins with a survey on transcendence and transcendental numbers, it then presents several properties of the transcendental numbers e and π, and then it gives the proof of a new inequality for transcendental numbers. Also, in relationship with these topics, we study solutions to the Yang-Baxter equation from hyperbolic functions and from logical implication.

The magnetic field of the earth plays an important role in the ecosystem, and the magnetic field of celestial bodies is also important in the formation of cosmic large-scale structures, but the origin and evolution of the celestial magnetic field is still an unresolved mystery. Many hypotheses to explain the origin have been proposed, but there are some insurmountable difficulties for each one. At present, the theory widely accepted in scientific society is the dynamo model, it says that, the movement of magnetofluid inside celestial bodies, which can overcome the Ohmic dissipative effect and generate persistent weak electric current and macroscopic magnetic field. However, this model needs an initial seed magnetic field, and the true values of many physical parameters inside the celestial body are difficult to obtain, and there is no stable solution to the large range of fluid motion. These are all difficulties for the dynamo model. Furthermore, it is difficult for the dynamo to explain the correlation between the dipole magnetic field and angular momentum of a celestial body. In this paper, by calculating the interaction between spin of particles and gravity of celestial body according to Clifford algebra, we find that a rotational celestial body provides a field $\Omega^\alpha$ for spins, which is similar to the magnetic field of a dipole, and the spins of charged particles within the celestial body are arranged along the flux line of $\Omega^\alpha$, then a macroscopic magnetic field is induced. The calculation shows that the strength of $\Omega^\alpha$ is proportional to the angular momentum of the celestial body, which explains the correlation between the magnetic intensity and angular momentum. The results of this paper suggest that further study of the effects of internal variables such as density, velocity, pressure and temperature of a celestial body on $\Omega^\alpha$ may provide some new insights into the origin and evolution of celestial magnetic field.

In order to understand the hot deformation behavior of novel Cu/Al laminated composites, isothermal hot compression tests were conducted by Gleeble-1500D thermo-mechanical simulator. And the effect of bonding interface, deformation temperature and strain rate on the deformation behavior was analyzed. Results show that under the interface constraint effect, soft Al layer trends to flow synchronously with hard Cu layer. And further microstructure examinations indicate the cooperative deformation capability of Cu/Al composites increases with increasing stain rate and decreasing deformation temperature. Strain hardening exponent, calculated based on the true stress-true strain data, also proves the effect of deformation temperature and strain rate on the cooperative deformation behavior. Meanwhile, unique composites structure allows the Al matrix to exhibit the characteristic of dynamic recrystallization during the hot deformation process. Lastly, strain compensated Arrhenius-type constitutive equation was employed to describe the coupling effect of temperature, strain rate and strain on the flow stress.

An expansion method based on time fractional Sine-Gordon equation is implemented to construct some real and complex valued exact solutions to the Korteweg-de Vries and modified Korteweg-de Vries equation in time fractional forms. Compatible fractional traveling wave transform plays a key role to be able to apply homogeneous balance technique to set the predicted solution. The relation between trigonometric and hyperbolic functions based on fractional Sine-Gordon equation allows to form the exact solutions with multiplication of powers of hyperbolic functions.

Stress relaxation tests in cantilever bending were performed on the C7025 and C7035 alloys at 298K and 393K，respectively. The effect of stress-relief treatments on stress relaxation properties was investigated. The structural changes associated with the stress relaxation process were examined using transmission electron microscope. The stress relaxation curve fits well to empirical formula σ*=[Kˊln(t+α₀)+C]^-n for stress relaxation. The curves can be split into two stages. The stress relax quickly in the first stage, slows down in the second stage, and tends to be a certain limit value after a long time. The curve and microstructure reveals that the C7035 alloy has a lower rate of stress relaxation and a higher anti-stress relaxation capacity than the C7025. The first reason is that the movement of vacancies required by spinodal decomposition is inhibited, and the quantity of cobalt -containing vacancies decreases dramatically in the C7035 alloy. The other reason is that the precipitated phases became uniformly diffused in the C7035 alloy. The precipitate phase uniformly distributes in the grain boundaries and the matrix, during the relaxed condition, and thus the dislocations moving is blocked by the precipitates.

Matiyasevich's theorem states that there is no algorithm to decide whether or not a given Diophantine equation has a solution in non-negative integers. Smorynski's theorem states that the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable. We prove: (1) Smorynski's theorem easily follows from Matiyasevich's theorem, (2 ) Hilbert's Tenth Problem for Q has a negative solution if and only if the set of all Diophantine equations with a finite number of rational solutions is not recursively enumerable.

We consider a multitude of topics in mathematics where unification constructions play an important role: the Yang–Baxter equation and its modified version, Euler’s formula for dual numbers, means and their inequalities, topics in differential geometry, etc. It is interesting to observe that the idea of unification (unity and union) is also present in poetry. Moreover, Euler’s identity is a source of inspiration for the post-modern poets.

A concise proof of the Riemann Hypothesis is presented by clarifying the Hadamard product expansion over the zeta zeros, demonstrating that the Riemann Hypothesis is true.

The present work is concerned with the study of the generalized Langevin equation and its link to the physical theories of statistical mechanics and scale relativity. It is demonstrated that the form of the coefficients of Langevin equation depend critically on the assumption of continuity of the reconstructed trajectory. This in turn demands for the fluctuations of the diffusion term to be discontinuous in time. This paper further investigates the connection between the scale-relativistic and stochastic mechanics approaches, respectively, with the study of the Burgers equation, which in this case appears as a stochastic geodesic equation. By further demanding time reversibility of the drift the Langevin equation can also describe equivalent quantum-mechanical systems in a path-wise manner. The resulting statistical description obeys the Fokker-Plank equation of the probability density of the differential system, which can be readily estimated from Monte Carlo simulations of the random paths. Based on the Fokker-Plank formalism a new derivation of the transient probability densities is presented. Finally, stochastic simulations are compared to the theoretical results.

In the paper, a fully discrete compact difference scheme with $O(\tau^{2}+h^{4})$ precision is established by considering the numerical approximation of the one-dimensional Allen-Cahn equation. The numerical solutions satisfy discrete maximum principle under reasonable step ratio and time step constraint is proved. And the energy stability for the fully discrete scheme is investigated. An example is finally presented to show the effectiveness of scheme.

Many systems in nature and laboratories are far from equilibrium and exhibit significant fluc- tuations, invalidating the key assumptions of small fluctuations and short memory time in or near equilibrium. A full knowledge of Probability Distribution Functions (PDFs), especially time- dependent PDFs, becomes essential in understanding far-from-equilibrium processes. We consider a stochastic logistic model with multiplicative noise, which has gamma distributions as stationary PDFs. We numerically solve the transient relaxation problem, and show that as the strength of the stochastic noise increases the time-dependent PDFs increasingly deviate from gamma distributions. For sufficiently strong noise a transition occurs whereby the PDF never reaches a stationary state, but instead forms a peak that becomes ever more narrowly concentrated at the origin. The addition of an arbitrarily small amount of additive noise regularizes these solutions, and re-establishes the existence of stationary solutions. In addition to diagnostic quantities such as mean value, standard deviation, skewness and kurtosis, the transitions between different solutions are analyzed in terms of entropy and information length, the total number of statistically distinguishable states that a system passes through in time.

In this paper we investigate the optimal investment strategy for a defined contribution (DC) pension plan during the decumulation phrase which is risk-averse and pays close attention to inflation risk. The plan aims to maximize the expected constant relative risk aversion (CRRA) utility from the terminal wealth by investing the wealth in a financial market consisting of an inflation-indexed bond, an ordinary zero coupon bond and a risk-free asset. We derive the optimal investment strategy in closed-form using the dynamic programming approach by solving the corresponding Hamilton-Jacobi-Bellman (HJB) equation. Our theoretical and numerical results reveal that under some rational assumptions, an inflation-indexed bond do has significant advantage to hedge inflation risk.

Electrically charged particles such as Electrons and Protons carry electric, E, and magnetic, B, fields. In addition to these fields, Quantum Mechanics (QM) endows these particles with an `arcane and spooky' field --- the wavefunction. This wavefunction of QM is not only assumed to be separate but distinct from the electromagnetic field. We herein upend this view by demonstrating otherwise. That is, we demonstrate that the four components of the Dirac wavefunction, can be shown to not only be an intimate, but, a direct function of the electromagnetic field carried by the particle in question. Insofar as unity, depth in our understanding and insight into both Dirac and Maxwell's equations as major pillars of Modern Physics, we believe that this work may very well inch us one-step-closer to the truth.

A mathematical model is proposed for quantitative description of the outbreak of novel coronavirus COVID-19 Although the model is relatively simple, the comparison with the public data shows that an exact solution solution of the model (with the correctly-specified parameters) leads to the results, which are in good agreement with the measured data in China and Austria. Prediction of the total number of the COVID-19 cases is discussed and examples are presented using the measured data in Austria, France and Poland.

A quantum time-dependent spectrum analysis, or simply, quantum spectral analysis (QSA) is presented in this work, and it’s based on Schrödinger’s equation. In the classical world, it is named frequency in time (FIT), which is used here as a complement of the traditional frequency-dependent spectral analysis based on Fourier theory. Besides, FIT is a metric which assesses the impact of the flanks of a signal on its frequency spectrum - not taken into account by Fourier theory and let alone in real time. Even more, and unlike all derived tools from Fourier Theory (i.e., continuous, discrete, fast, short-time, fractional and quantum Fourier Transform, as well as, Gabor) FIT has the following advantages, among others: a) compact support with excellent energy output treatment, b) low computational cost, O(N) for signals and O(N2) for images, c) it does not have phase uncertainties (i.e., indeterminate phase for a magnitude = 0) as in the case of Discrete and Fast Fourier Transform (DFT, FFT, respectively). Finally, we can apply QSA to a quantum signal, that is, to a qubit stream in order to analyze it spectrally.

We investigate the exact travelling wave solutions of the Tzitzéica-Dodd–Bullough–Mikhailov (TDBM) and Tzitzéica-type (TT) equations, appear in nonlinear optics, using simple ansatz approach. We use the Painlevé and traveling wave transformations to convert the mentioned nonlinear equations into nonlinear ordinary differential equations. Then, the simple ansatz approach has been employed to procure wave solutions, in terms of bright, singular soliton and trigonometric functions, for the Tzitzéica-Dodd–Bullough–Mikhailov (TDBM) and Tzitzéica-type (TT) equations. The study shows that the simple ansatz approach is straightforward and easy to implement for solving nonlinear problems.

This paper inquires on the options pricing modeling using Artificial Neural Networks to price Apple(AAPL) European Call Options. Our model is based on the premise that Artificial Neural Networks can be used as functional approximators and can be used as an alternative to the numerical methods to some extent, for a faster and an efficient solution. This paper provides a neural network solution for two financial models, the BlackScholes-Merton model, and the calibrated-Heston Stochastic Volatility Model, we evaluate our predictions using the existing numerical solutions for the same, the analytic solution for the Black-Scholes equation, COS-Model for Heston’s Stochastic Volatility Model and Standard Heston-Quasi analytic formula. The aim of this study is to find a viable time-efficient alternative to existing quantitative models for option pricing.

This work - in two parts - tries to promote a new method in Mathematical Modeling, not ever used before, namely the use of the methods of Non-standard Analysis in Topoi. For the general mathematical theory of topoi (Topoi Theory) we refer to the book [10] . For non-standard analysis in the particular Boolean topos SET (the category of sets) we refer to the book [11]. This first part of the work is related, from an applicative point of view, mainly with Mathematical Neuroscience, more precisely a model of the human thinking, consciousness and subconsciousness. The connections between brain and mind will be also scketched, but a closer study also need some further topics to be studied in Part II: Artificial Intelligence, namely (Quantum) neural networks and (Quantum) Turing machine. However both parts (I. and II.) should be considered together, because they complement each other. The topoi model the intuitionistic logic (multi-valued) and have been used in Quantum Physics (see [9] and its references) while Non-Standard Analysis in SET (introduced by Abraham Robinson[31]) has been applied in Mathematical Economics (see [2] and its references); however, the combination was never used until now in Applied Mathematics. Even more, the Non-standard Analysis in Topoi is not too much studied from the point of view of Pure Mathematics either. One of the main objective of this work is to produce progress in this aria of Mathematics also, mainly for the SET-type topoi and exponential topoi, those topoi used in Quantum Physics and in which we need Non-stantard Analysis for our purposes. Based on the paper [13] we propose the logic of non-standard extensions in topoi as a model of the human thinking (based on infons), these theories representing top and very difficult results in Abstract Mathematics. We will analise the relation between brain and mind via the genetic-epigenetic interplay. More details will be provided in the Introduction, followed by a short presentation of the general theory, later connected with the (Quantum) Yang-Baxter equations in topoi. The connections between these equations and braid groups and knots, with intended applications in Neuroscience and Artificial Intelligence will be further analysed in Part II. Besides promoting a new idea of modeling, the present paper aims to promote the building of a Research Proposal for the European $''$Human Brain Project$''$ (https://www.humanbrainproject.eu/en/), and the NSF-Europe Program $''$Collaborative Research in Computational Neuroscience$''$ (https://www.nsf.gov/funding/pgm\_summ.jsp?pims\_id=5147), jointly with the new founded $''$National Centre for the Research of the Brain$''$ of the Romanian Academy of Science (https://acad.ro/centreAR/CNCC/CNCC.pdf, in Romanian). $''$Blue Brain Project$''$ (https://www.epfl.ch/research/domains/bluebrain/) could also be of big interest. We hope that the journals edited by MDPI (Axioms, in particular) and by the Simion Stoilow Institute of the Romanian Academy - Revue Roumaine de Mathematique Pures et Appliqu (http://imar.ro/journals/Revue\_Mathematique/home\_page.html) - among others - will host special issues related to the research subjects described in both parts of this work, starting with this one. For further details and some of the progress on this research subject, see www.ovidiufpasarescu.com.}

In light of Liu \emph{at el.}'s original works, this paper revisits the solution of Burgers's nonlinear equation $u_t=a(u_x)^2+bu_{xx} $. The study found two exact and explicit solutions for groups $G_4$ and $G_6$, as well as a general solution. A numerical simulation is carried out. In the appendix a Maple code is provided

Current-quark masses are compared to the rest masses allowed by the Helmholtz equation in a polar model. Within the uncertainty of the current u quark mass determination, the current quark mass coincides with the rest mass allowed by the Helmholtz equation in the polar model in accordance with the second root of the zero Neumann function. Current d quark mass coincides with the rest mass calculated in accordance with the third root of the Bessel zero function. On the basis of a comparison of these results with the results obtained earlier for ordinary real particles u and d quarks stability is discussed.

We propose a new theory beyond the standard model of elementary-particle physics. Employing the concept of a quantized spacetime, our theory demonstrates that the zero-point energy of the vacuum alone is sufficient to create all the fields, including gravity, the static electromagnetic field, and the weak and strong interactions. No serious undetermined parameters are assumed. Furthermore, the relations between the forces at the quantum-mechanics level is made clear. Using these relations, we quantize Einstein’s gravitational equation: Beginning with the zero-point energy of the vacuum, and after quantizing Newtonian gravity equation, we combine the energies of a static electromagnetic field and gravity in a quantum spacetime. Applying these results to the Einstein gravity equation, we substitute the energy density derived from the zero-point energy in addition to redefining differentials in a quantized spacetime. This is how we derive the quantized Einstein gravitational equation without assuming the existence of macroscopic masses. For the weak interaction, by considering plane-wave electron and the zero-point energy, we obtain a wavefunction that represents a β collapse. In this process, from a different point of view than Weinberg-Salam theory, we derive the masses of the W and Z bosons and the neutrino, and we calculate the radius and lifetime of the neutron. For the strong interaction, we previously reported an analytical theory for calculating the mass of a proton by providing a specific linear attractive potential obtained from the zero-point energy, which agrees well with the measurements. In the present study, we further calculate the strong interaction between two nucleons, i.e., the mass of the pi-meson. The resulting calculated quantities agree with the measurements, which verifies our proposed theory.

We prove: (1) the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable, (2) the set of all Diophantine equations which have at most finitely many solutions in positive integers is not recursively enumerable, (3) the set of all Diophantine equations which have at most finitely many integer solutions is not recursively enumerable, (4) analogous theorems hold for Diophantine equations D(x₁, …, x_p) = 0, where p ∈ N\{0} and for every i ∈ {1, …, p} the polynomial D(x₁, …, x_p) involves a monomial M with a non-zero coefficient such that x_i divides M, (5) the set of all Diophantine equations which have at most k variables (where k ≥ 9) and at most finitely many solutions in non-negative integers is not recursively enumerable.

In our previous papers [1,3], using only the concepts of the zero-point energy and quantized space–times, all the fields including gravity were explained. However, the previous papers had the following limitations: First, the concept of the quantized space-time must be experimentally confirmed. Second, we should clarify the meaning of the quantized Einstein’s gravity equation, which is derived in [1]. Moreover, in another paper [2], we succeeded in describing the neutrinos’ self-energy and their oscillations. However, this paper assumes the rest energy of 3-leptons in advance, which is why we needed to uncover the reason why leptons have 3-generations. As mentioned, using the concepts of the zero-point energy and quantized space–times, we derived the quantized Einstein’s gravity equation in our previous paper [1]. The paper provides an analytical solution of this equalized Einstein’s equation, which implies the conservation of angular momentum in terms of quantized space–times. Employing this solution and without the standard big bang model, a unique form of acceleration equation for the acceleration-expansion universe is derived. Moreover, the temperature of the cosmic microwave background (CMB) emission is also obtained. Further, this solution results in an analytical (not numerical) derivation of the gravity wave. Moreover, based on the configuration of quantized space–times in terms of both electric and magnetic fields, we analytically attempted to calculate every equation in terms of electromagnetic and gravity fields, using the solution of the quantized Einstein’s gravity equation. As a result of this theory, first the calculated acceleration and temperature of CMB emission agree with the measurements. Furthermore, the analytical solution of the quantized Einstein’s gravity equation resulted in all the laws of electromagnetic and gravity fields in addition to the analytically derived gravity wave, which agrees well with the recent measurements. Moreover, the calculations of the energies in the basic configuration of the quantized space–times resulted in all 3-leptons’ rest energies. Considering this basic configuration is uniformly distributed everywhere in the universe, we can conclude that τ-particles or static magnetic field energy derived from the basic configuration of the quantized space–times is the identity of dark energy, which also distributes uniformly in the universe.

The description of the microscopic world in quantum mechanics is very different from that in classical physics, and there are some points of view that are contrary to intuition and logic. The first is the loss of reality, the behavior of micro particles shows randomness and hopping. The second is the loss of certainty, the conjugate physical variables of a system cannot be determined synchronously, they satisfy the Heisenberg uncertainty principle. The third is the non-local correlation. The measurement of one particle in the quantum entanglement pair will change the state of the other entangled particle simultaneously. In this paper, some concepts related to quantum entanglement, such as EPR correlation, quantum entanglement correlation function, Bell's inequality and so on, are analyzed in detail. Analysis shows that the mystery and confusion in quantum theory may be caused by the logical problems in its basic framework. Bell's inequality is only a mathematical theorem, but its physical meaning is actually unclear. The Bell state of quantum entangled pair may not satisfy the dynamic equation of quantum theory, so it cannot describe the true state of microscopic particles. In this paper, the correct correlation functions of spin entanglement pair and photonic entanglement pair are strictly derived according to normal logic. Quantum theory is a more fundamental theory than classical mechanics, and they are not parallel relation in logic. However, there are still some unreasonable contents in the framework of quantum theory, which need to be improved. In order to disclose the real relationship between quantum theory and classical mechanics, we propose some experiments which show the wave-particle duality simultaneously and provide intuitionistic teaching materials for the new interpretation of quantum theory.

A probabilistic description is essential for understanding the dynamics of stochastic systems far from equilibrium. To compare different Probability Density Functions (PDFs), it is extremely useful to quantify the difference among different PDFs by assigning an appropriate metric to probability such that the distance increases with the difference between the two PDFs. This metric structure then provides a key link between stochastic processes and geometry. For a non-equilibrium process, we define an infinitesimal distance at any time by comparing two PDFs at times infinitesimally apart and sum these distances in time. The total distance along the trajectory of the system quantifies the total number of different states that the system undergoes in time and is called the information length. By using this concept, we investigate classical and quantum systems and demonstrate the utility of the information length as a unique Lagrangian diagnostic to quantify the information change as a system continuously evolves in time and to map out attractor structure. We further elucidate quantum effects (uncertainty relation) and the dual role of the width of PDF in quantum systems.

We introduce a class of logarithmic wave equation. We study the global existence of week solution for this class of equation. We deal with the initial boundary value problem of this class. Using the Galerkin method and the Gross logarithmic Sobolev inequality we establish the main theorem of existence of week solution for this class of equation arising from Q-Ball Dynamic in particular.

Applying a procedure similar to that of E.S. Bring, by using a 4^th degree Tschirnhaus transformation, it was possible to transform the Bring-Jerrard normal quintic (BJQ) equation into a De Moivre form (DMQ), so that it could be solved by radicals. The general solution by radicals of the De Moivre equations of any degree is presented. By the same procedure the BJSx (normal sextic) equation was taken to another one without the 2^nd, 4^th and 6^th terms which was transformed into a cubic (solvable) equation. By applying a 6^th degree Tschirnhaus transformation to the BJSp (normal septic) equation its binormal (without the 2^nd, 3^rd, 4^th and 5^th terms) form was obtained.

DGJ (Daftardar-Gejii-Jafaris) method is used to obtain numerical solution of the third order fractional differential equation. Providing the DGJ method converges, the approximate solution is a good and effective numerical result which is close to the exact solution or the exact solution. For this,the examples of the explaning the method are presented. The proposed method is implemented for the approximation solution of the third order nonlinear fractional partial differential equations. The method was shown to be unsuitable and inconsistent for an example of a nonlinear fractional partial differential equation depend on initial-boundary value conditions. The fact that these numerical results are not consistent can be explained by the fact that the method is not convergent.